Proof positive (and negative)

In my new book Harmonics, I refute what every other scientist, professor and engineer in the world says in claiming that symmetrical distortion, such as over-driving an amplifier creates only Odd harmonics. In fact, in my video I make that point with gusto, but I was wrong. When I did some more research I found that virtually all distortion creates both Odd and Even harmonics, but in the case of symmetrical distortion the Even harmonics are cancelled out. It was one of those epiphanies that I come up with occasionally when I have been thinking long and hard about something that bothers me. Remember what Even harmonics mean, there are two full cycles of the second harmonic in any wave that has a second harmonic, and four full cycles of the fourth harmonic in any wave that has a fourth harmonic, there are six full cycles of .... you get the idea. But when the distortion is symmetrical, half of those cycles are in one phase and in the other half cycle, they are in the opposite phase. And just like matter and anti-matter being combined, those Even harmonics will cancel out, minus the universe-wide annihilation if you believe everything you see in Star Trek. So what happens when you have symmetrical distortion is that the Odd harmonics propagate through just fine, but the Even harmonics cancel each other out. Sounds crazy, but is it true?

In the above graphic you can see the time domain and frequency domain plots of two waves that have been corrupted by identical distortion, one on the positive peak and the other on the negative peak. You can see that the two spectra are identical, so in what way do the two spectra differ? I mean, the spectra of two different waveforms must differ in some way, but it's pretty obvious these two are identical! But you are forgetting that when you do an FFT you get two parts, cosine and sine, and through a rectangular to polar conversion we get magnitude and phase. In fact, a lot of people get confused when looking at the spectrum of a signal with a negative DC offset because the magnitude is always positive! But the polarity of the offset is not reflected in the magnitude, but in the phase! Surprise surprise as Gomer Pyle would say, a spectrum contains two quantities, and after processing we get magnitude and phase.

Above you can see the phases of the two kinds of distorted waves for harmonics. The Positive Peak Clipped shows the phases of the Even harmonics from the spectrum (remember, two parts to a spectrum, magnitude and phase) of the wave at the top-left of the graphic at the top of this article and the Negative Peak Clipped shows the phases of the Even harmonics from the spectrum of the wave at the top-right of the graphic at the top of this article. You will notice that the first entry in the table confirms what I said earlier about DC offset. For the Positive Peak Clipped, the phase is 180 degrees, the Negative Peak column has a phase of 0 degrees. That may seem counter intuitive but remember that when I chop the top off of a sine wave, the DC offset goes below zero, making the DC offset (average value) negative. When I chop the bottom off a sine wave, the DC offset goes positive, so the phase is zero. Please note that these exact values only apply to Excel's FFT, other FFTs may have different polarity.

Then note that we get this interesting pattern in the middle (Positive) column, 180, 0, 180, 0, 180, 0, 0, 180, 0, 180, 180, and so on. But what is of most interest here is that the column next to it is exactly the opposite! 0, 180, 0, 180, 0, 180, 180, 0, 180, 0, 0. So when I chop the top off a sine wave, the Even harmonics have one set of phases, but when I chop the bottom off a sine wave, the Even harmonics have the opposite polarity, exactly 180 degrees out of phase. So what do you think happens when I chop both the top and bottom off a sine wave? You should already know the answer: The Even harmonics cancel out! For half the wave, you have Even harmonics in one polarity, and in the other half of the wave you have Even harmonics in the opposite polarity. What's going to happen? They are going to cancel out, just like those fancy noise cancelling headphones take the ambient noise, reverse the phase and add it into the audio stream going to the headphones, canceling the external noise. Same thing here.

So, I'm sorry to say that everybody out there, including Audio Precision is dead wrong. I was wrong in my video, too. It's not that symmetrical distortion creates only Odd harmonics. In my video what I am showing you is how you can balance the clipping waveform so that the positive peak and the negative peak are perfectly balanced, allowing the Even harmonics to perfectly cancel each other out. In my defense I will say that the whole reason I made the video was because I knew of this phenomenon, but didn't know exactly why it happened. But now I know, and now you do, too.